cp's OEIS Frontend

A094572 Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.

%I A094572 #17 Jan 09 2025 19:08:16
%S A094572 2,0,4,2,4,0,4,4,6,0,4,4,4,0,8,6,4,0,4,4,8,0,4,8,6,0,8,4,4,0,4,8,8,0,
%T A094572 8,6,4,0,8,8,4,0,4,4,12,0,4,12,6,0,8,4,4,0,8,8,8,0,4,8,4,0,12,10,8,0,
%U A094572 4,4,8,0,4,12,4,0,12,4,8,0,4,12,10,0,4,8,8,0,8,8,4,0,8,4,8,0,8,16,4,0,12,6
%N A094572 Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.
%C A094572 The old entry with this sequence number was a duplicate of A058071.
%C A094572 a(n) == 2 (mod 4) if n is a square otherwise a(n) is divisible by 4. Cf. A112329. - _Peter Bala_, Jan 08 2025
%D A094572 M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
%H A094572 Ray Chandler, <a href="/A094572/b094572.txt">Table of n, a(n) for n = 1..10000</a>
%H A094572 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references).
%F A094572 a(n) = 2*d(n) if n is odd, = 2*d(n/4) if n == 0 mod 4, otherwise 0, where d() = A000005().
%F A094572 a(n) = 2 * A112329(n). - _Ray Chandler_, Aug 23 2014
%F A094572 From _Amiram Eldar_, Apr 13 2024: (Start)
%F A094572 Dirichlet g.f.: 2*zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s)).
%F A094572 Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma-1)*n, where gamma is Euler's constant (A001620). (End)
%p A094572 with(numtheory); f:=proc(n) if n mod 2 = 1 then RETURN(2*tau(n)); fi; if n mod 4 = 0 then RETURN(2*tau(n/4)); fi; 0; end;
%t A094572 Table[If[OddQ[n],2DivisorSigma[0,n],If[OddQ[n/2],0,2DivisorSigma[0,n/4]]],{n,100}] (* _Ray Chandler_, Aug 23 2014 *)
%o A094572 (PARI) a(n) = if(n%2, 2 * numdiv(n), if(n % 4 == 0, 2 * numdiv(n/4), 0)); \\ _Amiram Eldar_, Apr 13 2024
%Y A094572 Cf. A000005, A001620, A093061, A112329.
%K A094572 nonn,easy
%O A094572 1,1
%A A094572 _N. J. A. Sloane_, Nov 02 2008